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Normal matrices

Created over 8 years ago, updated 10 days ago

Definition: A complex square matrix $A$ is normal if it commutes with its conjugate transpose:

$$AA^* = A^*A$$

Example: All of the following are normal matrices:

  • Hermitian matrices ($A = A^*$)
  • Skew-Hermitian matrices ($A = -A^*$)
  • Unitary matrices ($A^* = A^{-1}$)
  • Symmetric matrices ($A = A^T$)
  • Orthogonal matrices ($A^T = A^{-1}$)

Spectral Theorem for Normal Matrices: $A$ is normal if and only if $A$ is unitarily diagonalizable: $A = UDU^*$ where $D$ is diagonal and $U$ is unitary.

Properties:

  • Normal matrices have an orthonormal basis of eigenvectors
  • The sum and product of commuting normal matrices is normal
  • If $A$ is normal, then $\|Ax\| = \|A^*x\|$ for all $x$

Normal matrices generalize Hermitian and unitary matrices, capturing exactly those matrices that can be diagonalized by a unitary transformation.